Relative Rain

 

If someone goes from his car to his front door in a rainstorm, will he get more wet, less wet, or equally wet if he runs rather than walks? To develop a quantitative answer, consider a spherical man, and assume he moves to his car in a straight horizontal path with velocity u. The raindrops are falling at an angle such that their velocity is vz in the downward direction, vx in the horizontal direction (straight into the man's face), and vy in the sideways horizontal direction (the man's left to right). The intensity of the rain is such that each cubic foot of air contains r grams of water.

 

Relative to the rain's frame of reference the raindrops are stationary and the man and his car both have an upward velocity vz and a sideways velocity (right to left) of vy. In addition, the man has a forward horizontal velocity of u + vx. Clearly the amount of rain encountered by the man is equal to r times the volume of space he sweeps out as he moves relative to this stationary mist of raindrops. Since he is spherical with radius R, this swept volume is essentially equal to pR2L, where L is the distance traveled (relative to the rain's frame of reference).

 

If D is the horizontal distance to the car (in the frame of the ground), and the man moves straight to his car with velocity u, the time it takes him is D/u. His total velocity relative to the falling rain is

 

 

so the distance he moves relative to the rain is L = (D/u)Vt. Therefore, the amount of rain he encounters in the general case for arbitrary direction of rainfall is

 

 

We can easily incorporate other assumptions, such as the man having some non-spherical shape. It's just a matter of geometry to compute how much volume he sweeps out relative to the rain's frame of reference. This is done by replacing pR2 with the horizontal facing cross-sectional area A of the man in terms of the rest frame of the rain.

 

If vx = vy = 0 then the rain is falling vertically with a total velocity v = vz. In this case equation (2) reduces to

 

 

which shows that the key parameter is the ratio of the rain's vertical speed to the man's horizontal speed. Of course, if v was zero (which would mean the rain was motionless relative to the ground), then L would always equal D, and W would equal rAD, regardless of how fast the man runs. On the other hand, for any v greater than zero, the amount of rain he encounters will go down as his horizontal velocity u increases. Of course, in this case, if the man is not moving at all (i.e., u = 0) he will get infinitely wet.

 

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